A conjecture about Dirichlet series.

[tpm]

if $$g(s)= \sum_{n=1}^{\infty} a(n) n^{-s}$$

Where g(s) has a single pole at s=1 with residue C, then my question/conjecture is if for s >0 (real part of s bigger than 0) we can write

$$g(s)= C(\frac{1}{s-1}+1)-s\int_{0}^{\infty}dx(Cx-A(x))x^{-s-1}$$

of course $$A(x)=\sum_{n \le x}a(n)$$

the question is if the series converge for s >1 with a pole there is a method to 'substract' this singularity (pole) at s=1 to give meaning for the series at any positive s.

I think that the 'Ramanujan resummation' may help to give the result:

$$\sum_{ n >1}^{[R]}a(n)n^{-s} = g(s)-C(s-1)^{-1}$$ valid even for s=1 or s>0 (??)

 

[DeadWolfe]

Jose, how's it been?

 

[tacman]

Your conjecture is interesting and has potential implications for the study of Dirichlet series. However, it is important to note that this is just a conjecture and would need to be rigorously proven before it can be accepted as a valid result. It would also be helpful to have some examples or specific cases where this conjecture holds true, as it would provide evidence for its validity. Additionally, further exploration and analysis of Ramanujan resummation and its applications in this context could potentially strengthen your conjecture. Overall, this is a thought-provoking idea and could potentially contribute to the understanding of Dirichlet series.